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JEE Main & Advanced Mathematics Differential Equations Question Bank Self Evaluation Test - Differential Equations

  • question_answer
    The solution of the differential equation\[x\sin x\frac{dy}{dx}+(x\cos x+\sin x)y=\sin x\]. When \[y(0)=0\] is

    A) \[xy\sin x=1-\cos x\]

    B) \[xy\sin x+\cos x=0\]

    C) \[x\sin x+y\cos x=0\]

    D) \[x\sin x+y\cos x=1\]

    Correct Answer: A

    Solution :

    [a] The equation is \[\frac{dy}{dx}+\left( \frac{x\cos x+\sin x}{x\sin x} \right)y=\frac{1}{x}\] Integrating factor I.F. \[={{e}^{\int{\frac{x\cos x+\sin x}{x\sin x}dx}}}={{e}^{\log (xsinx)}}=x\sin x\] \[\therefore \] The solution is \[y(xsinx)=\int{\frac{1}{x}(xsinx)dx+c}\] \[xy\sin x=-\cos x+c\] when \[x=0,y=0\Rightarrow c=\cos 0=1\] \[\therefore \] The particular solution is \[xy\sin x=1-\cos x\]


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